Geosci. Model Dev., 7, 1051–1068, 2014
www.geosci-model-dev.net/7/1051/2014/
doi:10.5194/gmd-7-1051-2014
© Author(s) 2014. CC Attribution 3.0 License.
Earth Orbit v2.1: a 3-D visualization and analysis model of Earth’s
orbit, Milankovitch cycles and insolation
T. S. Kostadinov1,2,* and R. Gilb1
1 Department
2 Formerly
of Geography and the Environment, 28 Westhampton Way, University of Richmond, Richmond, VA 23173, USA
at Earth Research Institute, University of California Santa Barbara, Santa Barbara, CA 93106, USA
* T.
S. Kostadinov dedicates this paper to his mother, who first sparked his interest in the magnificent night sky and the science
of astronomy, and to his father, who showed him the pivotal importance of mathematics.
Correspondence to: T. S. Kostadinov ([email protected])
Received: 4 October 2013 – Published in Geosci. Model Dev. Discuss.: 28 November 2013
Revised: 25 March 2014 – Accepted: 10 April 2014 – Published: 3 June 2014
Abstract. Milankovitch theory postulates that periodic variability of Earth’s orbital elements is a major climate forcing
mechanism, causing, for example, the contemporary glacial–
interglacial cycles. There are three Milankovitch orbital parameters: orbital eccentricity, precession and obliquity. The
interaction of the amplitudes, periods and phases of these parameters controls the spatio-temporal patterns of incoming
solar radiation (insolation) and the timing and duration of
the seasons. This complexity makes Earth–Sun geometry and
Milankovitch theory difficult to teach effectively. Here, we
present “Earth Orbit v2.1”: an astronomically precise and accurate model that offers 3-D visualizations of Earth’s orbital
geometry, Milankovitch parameters and the ensuing insolation forcing. The model is developed in MATLAB® as a userfriendly graphical user interface. Users are presented with a
choice between the Berger (1978a) and Laskar et al. (2004)
astronomical solutions for eccentricity, obliquity and precession. A “demo” mode is also available, which allows the
Milankovitch parameters to be varied independently of each
other, so that users can isolate the effects of each parameter
on orbital geometry, the seasons, and insolation. A 3-D orbital configuration plot, as well as various surface and line
plots of insolation and insolation anomalies on various time
and space scales are produced. Insolation computations use
the model’s own orbital geometry with no additional a priori
input other than the Milankovitch parameter solutions. Insolation output and the underlying solar declination computation are successfully validated against the results of Laskar et
al. (2004) and Meeus (1998), respectively. The model outputs
some ancillary parameters as well, e.g., Earth’s radius-vector
length, solar declination and day length for the chosen date
and latitude. Time-series plots of the Milankovitch parameters and several relevant paleoclimatological data sets can
be produced. Both research and pedagogical applications are
envisioned for the model.
1
Introduction
The astrophysical characteristics of our star, the Sun, determine to first order the continuously habitable zone around it
(Kasting et al., 1993; Kasting, 2010), in which rocky planets
are able to maintain liquid water on their surface and sustain life. The surface temperature of a planet depends to first
order upon the incoming flux of solar radiation (insolation)
to its surface. Additionally, energy for our metabolism (and
most modern economic activities) is obtained exclusively
from the Sun via the process of oxygenic photosynthesis performed by green terrestrial plants and marine phytoplankton.
The high oxygen content of Earth’s atmosphere, necessary
for the evolution of placental mammals (Falkowski et al.,
2005), is due to billions of years of photosynthesis and the
geological burial of reduced carbon equivalents (Falkowski
and Godfrey, 2008; Falkowski and Isozaki, 2008; Kump et
al., 2010). Thus, the Sun is central to climate formation and
stability and to our evolution and continued existence as a
species.
Published by Copernicus Publications on behalf of the European Geosciences Union.
1052
The temporal and spatial patterns of insolation and their
variability on various scales determine climatic stability over
geologic time, as well as climate characteristics such as diurnal, seasonal and pole to equator temperature contrasts, all of
which influence planetary habitability. Insolation can change
due to changes in the luminosity of the Sun itself. This can
happen due to the slow increase of solar luminosity that gives
rise to the faint young Sun paradox (Kasting, 2010; Kump et
al., 2010), or it can happen on much shorter timescales such
as the 11-year sunspot cycle (Fröhlich, 2013; Hansen et al.,
2013).
Importantly, insolation is also affected by the orbital elements of the planet. According to the astronomical theory of
climate, quasi-periodic variations in Earth’s orbital elements
cause multi-millennial variability in the spatio-temporal distributions of insolation, and thus provide an external forcing
and pacing to Earth’s climate (Milankovitch, 1941; Berger
1988; Berger and Loutre, 1994; Berger et al., 2005). These
periodic orbital fluctuations are called Milankovitch cycles,
after the Serbian mathematician Milutin Milanković who was
instrumental in developing the theory (Milankovitch, 1941).
Laskar et al. (2004) provide a brief historical overview of
the main contributions leading to the pioneering work of
Milanković. There are three Milankovitch orbital parameters: orbital eccentricity (main periodicities of ∼ 100 and
400 kyr (1 kyr = one thousand years)), precession (quantified
as the longitude of perihelion relative to the moving vernal
equinox, main periodicities ∼ 19 and 23 kyr) and obliquity of
the ecliptic (main periodicity 41 kyr) (Berger, 1978a). Obliquity is strictly speaking a rotational, rather than an orbital
parameter; however, we refer to it here either as an orbital or
Milankovitch parameter, for brevity.
The pioneering work by Hays et al. (1976) demonstrated
a strong correlation between these cycles and paleoclimatological records. Since then, multiple analyses of paleoclimate
records have been found to be consistent with Milankovitch
forcing (e.g., Imbrie et al., 1992; Rial, 1999; Lisiecki and
Raymo, 2005). Notably, the glacial–interglacial cycles of
the Quaternary have been strongly linked to orbital forcing, particularly summertime insolation at high northern latitudes (Milankovitch, 1941; Berger 1988; Berger and Loutre,
1994; Bradley, 2014, and references therein). Predicting the
Earth system response to orbital forcing (including glacier
growth and melting) is not trivial, and there are challenges in
determining which insolation quantity (i.e., integrated over
what time and space scales) is responsible for paleoclimate
change, e.g., peak summer insolation intensity, or overall
summertime-integrated insolation at northern latitudes (Imbrie et al., 1993; Lisiecki et al., 2008; Huybers, 2006; Huybers and Denton, 2008; Bradley, 2014). Moreover, some controversies related to the astronomical theory remain, notably
the 100 kyr problem, or the so-called mid-Pleistocene transition. This refers the fact that the geological record indicates
that the last ca. one million years have been dominated by
100 kyr glacial–interglacial cycles, a gradual switch from the
Geosci. Model Dev., 7, 1051–1068, 2014
T. S. Kostadinov and R. Gilb: Earth Orbit v2.1
previously dominant 41 kyr periodicity. This transition cannot be explained by orbital forcing alone, as there was actually a decrease in the 100 kyr eccentricity band variance
during this period (e.g., Imbrie et al., 1993; Loutre et al.,
2004; Berger et al., 2005; Bradley, 2014, and references
therein). Current consensus focuses on the explanation that
the mid-Pleistocene transition is due to factors within the
Earth system itself, rather than astronomical factors – e.g.,
internal climate system oscillations, nonlinear responses due
to the continental ice sheet size, or CO2 degassing from the
Southern Ocean (Bradley, 2014, Sects. 6.3.3 and 6.3.4, and
references therein). Finally, alternative astronomical influences on climate have also been proposed, such as the influence of the orbital inclination cycle (Muller and McDonald,
1997).
The Milankovitch cycles are due to complex gravitational
interactions between the bodies of the solar system. Astronomical solutions for the values of the Milankovitch orbital parameters have been derived by Berger (1978a) and
Berger (1978b), referred to henceforth as Be78 (valid for
1000 kyr before and after present), and Laskar et al. (2004),
referred to henceforth as La2004 (valid for 101 000 kyr before present to 21 000 kyr after present). Here the present is
defined as the start of Julian epoch 2000 (J2000), i.e., the
Gregorian calendar date of 1 January, 2000 at 12:00 UT (Universal Time, that is, mean solar time at the Prime Meridian + 12 h) (Meeus, 1998). There are several other solutions
as well, for example, Berger and Loutre (1992) and Laskar et
al. (2011). These astronomical solutions are crucial for paleoclimate and climate science, as they enable the computation
of insolation at any latitude and time period in the past or
future within the years spanned by the solutions (Berger and
Loutre, 1994; Berger et al., 2010; Laskar et al., 2004), and
subsequently the use of this insolation in climate models as
forcing (e.g., Berger et al., 1998). Climate models are an important method for testing the response of the Earth system
to Milankovitch forcing.
While most Earth science students and professionals are
well aware of Earth’s orbital configuration and the basics
of the Milankovitch cycles, the details of both and the way
the Milankovitch orbital elements influence spatio-temporal
patterns of insolation on various time and space scales remain elusive. It is difficult to appreciate the pivotal importance of Kepler’s laws of planetary motion in controlling the
effects of Milankovitch cycles on insolation patterns. The
three-dimensional nature of Earth’s orbit, the vast range of
space and timescales involved, and the geometric details are
complex, and yet those same factors present themselves to
computer modeling and 3-D visualization. Here, we present
“Earth Orbit v2.1”: an astronomically precise and accurate
3-D visualization and analysis model of Earth’s orbit, Milankovitch cycles, and insolation. The model is envisioned
for both research and pedagogical applications and offers 3D visualizations of Earth’s orbital geometry, Milankovitch
parameters and the ensuing insolation forcing. It is developed
www.geosci-model-dev.net/7/1051/2014/
T. S. Kostadinov and R. Gilb: Earth Orbit v2.1
in MATLAB® and has an intuitive, user-friendly graphical
user interface (GUI) (Fig. 1). Users are presented with a
choice between the Be78 and La2004 astronomical solutions
for eccentricity, obliquity and precession. A “demo” mode
is also available, which allows the three Milankovitch parameters to be varied independently of each other (and exaggerated over much larger ranges than the naturally occurring
ones), so users can isolate the effects of each parameter on orbital geometry, the seasons, and insolation. Users select a calendar date and the Earth is placed in its orbit using Kepler’s
laws; the calendar can be started on either vernal equinox
(20 March) or perihelion (3 January). A 3-D orbital configuration visualization, as well as spatio-temporal surface and
line plots of insolation and insolation anomalies (with respect
to J2000) on various scales are then produced. Below, we first
describe the model parameters and implementation. We then
detail the model user interface, provide instructions on its capabilities and use, and describe the output. We then present
successful model validation results, which are comparisons
to existing independently derived insolation, solar declination and season duration values. Finally, we conclude with
brief analysis of sources of uncertainty. Throughout, we provide examples of the pedagogical value of the model.
Various insolation solutions and visualizations exist
(Berger, 1978a; Rubincam, 1994 (however, see response of
Berger, 1996); Laskar et al., 2004; Archer, 2013; Huybers,
2006). Notably, the AnalySeries software (Paillard et al.,
1996; Paillard, 2014) shares many of the functionalities presented here and offers many additional ones, such as paleoclimatic time-series analysis and many more choices for insolation computation. Importantly, the model presented here
was developed independently from AnalySeries (or other
similar efforts) and computes insolation from first principles
of orbital mechanics (Kepler’s laws) and irradiance propagation, using exclusively its own internal geometry. The only
model inputs are the three Milankovitch orbital parameters,
either real astronomical solutions (Be78 or La2004) or userentered demo values. No insolation computation code from
the above-cited existing solutions has been used, so comparison with these solutions constitutes independent model verification, referred to here as validation, because we consider
the La2004 and Meeus (1998) solutions the geophysical truth
(Sect. 5).
The unique contribution of our model consists of the combination of the following features:
a. central to the whole model is a user-controllable, 3-D
pan–tilt–zoom plot of the actual Earth orbit;
b. an interactive user-friendly GUI that serves as a singleentry control panel for the entire model and makes it
suitable for use by non-programmers and friendly to didactic applications;
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c. the Milankovitch cycles are incorporated explicitly and
insolation is output according to real or user-selected
demo orbital elements, which
d. allows users to enter exaggerated orbital parameters independently of each other and isolate their effects on
insolation, as well as view the orbit with exaggerated
eccentricity;
e. the source code is published and advanced users can
check its logic, as well as modify it and adapt it; and
f. the software is platform-independent.
The issue of climate change has come to the forefront of
Earth science and policy and it is arguably the most important
global issue of immediate and long-term consequences (e.g.,
IPCC, 2013). Earth’s climate varies naturally over multiple
timescales, from decadal to hundreds of millions of years
(e.g., Kump et al., 2010). It is thus crucial to understand natural climate forcings, their timescales, and the ensuing response of the Earth system. In addition, detailed understanding of the Sun’s daily path in the sky and the patterns of insolation have become important to increasing numbers of students and professionals because of the rise in usage of solar
power (thermal and photovoltaic). We submit that the model
presented here can enhance understanding of all of these important subject areas.
2
Key definitions, model parameters and
implementation
The model input parameters, and their values and units, are
summarized in Table 1. The following definitions, discussion and symbols are consistent with those of Berger et
al. (2010). The reader is referred to their Fig. 1. According to Kepler’s first law of planetary motion, Earth’s orbit
is an ellipse, and the Sun is in one of its foci (e.g., Meeus,
1998). Orbital eccentricity, e (Table 1), is a measure of the
deviation of p
Earth’s orbital ellipse from a circle and is defined as e = 1 − b2 /a 2 , where a is the semi-major axis (Table 1) and b is the semi-minor axis of the orbital ellipse (e.g.,
Berger and Loutre, 1994). The semi-major axis is equal to
about 1 AU (Meeus, 1998; Standish et al., 1992) and determines the size of the orbital ellipse and thus the orbital period
of Earth; it is considered a fixed constant in the model, as its
variations are extremely small (Berger et al., 2010; Laskar
et al., 2004, their Fig. 11). Various orbital period definitions
are possible; here, the sidereal period is used as a model constant (Meeus, 1998). Thus, Kepler’s third law of planetary
motion is implicit in these two constant definitions and is
not included explicitly elsewhere in model logic. The obliquity of the ecliptic, ε, is the angle between the direction of
Earth’s axis of rotation and the normal to the orbital plane, or
the ecliptic (Table 1). Eccentricity and obliquity are two of
the three Milankovitch orbital parameters.
Geosci. Model Dev., 7, 1051–1068, 2014
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T. S. Kostadinov and R. Gilb: Earth Orbit v2.1
Figure 1. Main MATLAB® GUI window of Earth Orbit v2.1. Input and output displayed corresponds to the graphical output of Fig. 2a, i.e.,
contemporary (J2000) La2004 configuration for 16 September, at 43◦ N latitude.
The third Milankovitch orbital parameter, precession, is
the most challenging for instruction and visualization. There
are two separate kinds of precession that combine to create
a climatic effect – precession of the equinoxes (also termed
axial precession), and apsidal precession, that is, precession
of the perihelion in the case of Earth’s orbit. Axial precession refers to the wobbling of Earth’s axis of rotation that
slowly changes its absolute orientation in space with respect to the distant stars. The axis or rotation describes a
cone (one in each hemisphere) in space with a periodicity
of about 26 000 years (Berger and Loutre, 1994). This is
the reason why the star α UMi (present-day Polaris, or the
North Star), has not and will not always be aligned with the
direction of the North Pole. Also, due to axial precession,
the point of vernal equinox in the sky moves with respect to
the distant stars and occurs in successively earlier zodiacal
Geosci. Model Dev., 7, 1051–1068, 2014
constellations. Axial precession is clockwise as viewed from
above the North Pole, hence the north celestial pole describes
a counterclockwise motion as viewed by an observer looking in the direction of the north ecliptic pole. Precession
of the perihelion refers to the gradual rotation of the line
joining aphelion and perihelion, with respect to the distant
stars (or the reference equinox of a given epoch) (Berger,
1978a; Berger and Loutre, 1994).
Axial precession and precession of the perihelion combine to modulate the relative position of the equinoxes and
solstices (i.e., the seasons) with respect to perihelion, which
is what is relevant for insolation and climate. This climatically relevant precession is implemented in the model and
is quantified via the longitude of perihelion, ω̃, which is
the angle between the directions of the moving fall equinox
and perihelion at a given time, measured counterclockwise
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Table 1. Summary of constant and variable model input parameters.
Symbol
Constant/variable
Value
Units
Reference
Notes
106 km
constant
1.00000261 AU
(constant)
constant
AU
a
Astronomical unit
Semi-major axis
149.597870700
149.598261150
106 km
USNO (2013)
Standish et al. (1992)
T
365.256363
Days
Meeus (1998)
So
Sidereal orbital
period
TSI at 1 AU
1366a
W m−2
Fröhlich (2013)
e
ε
ω̃
Eccentricity
Obliquity
Longitude of perihelion
0.01670236b
23.4393b
102.9179b
–
degrees
degrees
La2004c
La2004
La2004
Also see Kopp
and Lean (2011)
–
–
–
a Users can change this default value. b Default J2000 values. Users can change these variables independently of each other or choose real
astronomical solutions depending on the mode selected. c La2004 refers to Laskar et al. (2004)
in the plane of the ecliptic (Berger et al., 2010). Because
both perihelion and equinox move, the longitude of perihelion will have a different (shorter) periodicity than one full
cycle of axial wobbling alone (Berger and Loutre, 1994).
The direction of Earth’s radius vector when Earth is at fall
equinox (∼ 22 September) is referred to as the direction of
fall equinox above. This is the direction with respect to the
distant stars where the Sun would be found on its annual motion on the ecliptic on 20 March – that is, at vernal equinox.
In other words, that is the direction of the vernal point in
the sky (Berger et al., 2010; their Fig. 1 and Appendix B),
the origin of the right ascension coordinate. This distinction between vernal equinox and the direction of the vernal
point can cause confusion, especially since the exact definition of longitude of perihelion can vary (e.g., c.f. Berger,
1978a; Berger et al., 1993, 2010; Berger and Loutre, 1994;
Joussaume and Braconnot, 1997) and the longitude of perihelion can also be confused with the longitude of perigee,
ω = ω̃ + 180◦ , which is the angle between the directions of
vernal equinox and perihelion, measured counterclockwise
as viewed from the direction of the North Pole, in the plane
of the orbit (Berger et al., 2010). Here, we use the terminology and definitions of Berger et al. (2010).
The magnitude of the climatic effect of precession is modulated by eccentricity. In the extreme example, if eccentricity
were exactly zero, the effects of precession would be null.
Climatic precession, e sin ω, is the parameter that quantifies
precession and determines season lengths, the Earth–Sun distance at summer solstice (Berger and Loutre, 1994) and various important insolation quantities (Berger et al., 1993, their
Table 1). This interplay between eccentricity and precession
presents an important way to introduce both concepts pedagogically and to test student comprehension.
The solar “constant”, So , is defined here as the total solar irradiance (TSI) on a flat surface perpendicular to the solar rays at a reference distance of exactly 1 AU (Table 1).
As Berger et al. (2010) note, due to eccentricity changes,
the mean distance from the Earth to the Sun over a year is
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not constant on geologic timescales. It also matters how this
mean distance is defined – for example, over time (mean
anomaly) vs. over angle (true anomaly). True and mean
anomaly are defined below in Sects. 2.1 and 2.2, respectively. If So is defined to be the irradiance from the Sun at
the mean Earth–Sun distance, then it is indeed not a true
constant. As used here, So is a true model constant as long
as the luminosity of the Sun itself is assumed constant. The
default value is chosen to be 1366 W m−2 (Fröhlich, 2013).
Recent evidence suggests that the appropriate value may actually be about 1361 W m−2 (Kopp and Lean, 2011). Users
can change the value of So independently of other model inputs in order to study the effects of changes in absolute solar
luminosity – for example, in order to simulate the faint young
Sun (e.g., Kasting, 2010) or the sunspot cycle (e.g., Hansen
et al., 2013).
2.1
Model coordinate system; Sun–Earth geometry
parameterization; solar declination
According to Kepler’s first law of planetary motion, Earth orbits the Sun in an ellipse, and the Sun is in one of the ellipse’s
foci. The heliocentric equation of the orbital ellipse in polar
form is given by (Meeus, 1998; his Eq. 30.3)
|r(ν)| =
a(1 − e2 )
.
1 + e cos ν
(1)
In the above, the Sun is at the origin of the coordinate system; a is the semi-major axis of the orbital ellipse; e is eccentricity; ν is true anomaly; and r is Earth’s instantaneous radius vector, that is, the vector originating at the Sun and ending at the instantaneous planetary position. Let r designate
the length of Earth’s radius vector henceforth. True anomaly,
ν, is the angle between the directions of perihelion and the
radius vector, subtended at the Sun and measured counterclockwise in the plane of the orbit (e.g., Meeus, 1998, his
Ch. 30; Berger et al., 2010, their Fig. 1). The true longitude
of the Sun (or simply true longitude) is equal to Earth’s true
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anomaly plus the longitude of perigee (Berger et al., 2010,
their Eq. 6). True longitude is the angle Earth has swept from
its orbit, subtended at the Sun, since it was last at vernal
equinox, and it is equivalent to the angle the Sun has travelled along the ecliptic in the same time. Mean longitude is
the longitude of the mean Sun, in an imaginary perfectly circular orbit of the same period, that is, mean longitude is proportional to the passage of time, much like mean anomaly
(see Sect. 2.2 below).
In the Earth Orbit v2.1 model, given a user-selected calendar date, true anomaly, ν, is determined by solving the inverse Kepler equation (see Sect. 2.2 below). The Earth’s radius vector is then solved for using Eq. (1) above. Because
the main model coordinate system is heliocentric Cartesian,
the (r, ν) pair of polar coordinates is then transformed to
Cartesian (x, y) for plotting. The model’s main coordinate
system has its origin at the center of the Sun, its positive
x axis pointing in the direction of perihelion, its positive
y axis pointing 90 degrees counterclockwise in the plane of
the ecliptic, and its positive z axis perpendicular to it in the
direction of the north ecliptic pole. The Earth is initially parameterized as a sphere in its own geocentric Cartesian coordinate system in terms of its radius and geographic latitude
and longitude (corresponding to the two angles of a spherical
coordinate system). The Earth’s coordinate system’s x and y
axes are in the plane of the Equator (shown as a black dotted line, Fig. 2), and its z axis is pointing towards the true
North Pole and is coinciding with Earth’s axis of rotation;
these axes are also plotted in black dotted lines; the z axis is
lengthened so that it pierces Earth’s surface at the Poles, and
the North Pole is labeled. Earth is plotted as a transparent
mesh so that important orbital elements can be seen through
it at various zoom levels (Fig. 2). The color scale of Earth’s
mesh is just a function of latitude and no day and night sides
are explicitly shown. Earth’s radius is not to scale with the
orbit itself or with the Sun’s radius. Thus, the center of Earth
has its true geometric orbital position (and is the tip of its instantaneous radius vector); however, the surface of the sphere
in the model is arbitrary and must not be interpreted as the
true surface onto which insolation is computed, for example.
The insolation computations (Sect. 2.3) are geocentric. The
Sun is also plotted (not to scale) as a sphere centered at the
origin of the main model coordinate system.
The Earth is oriented properly in 3-D with respect to the
orbital ellipse by using a rotation matrix to rotate its coordinate system. The 3-D rotation matrix is computed using Rodrigues’ formula (Belongie, 2013) for 3-D rotation
about a given direction by a given angle. The direction about
which Earth is rotated is determined by a vector which is
always in the orbital plane (k component is zero), and the
i and j components are determined by the longitude of
perihelion. The angle by which Earth is rotated is determined by obliquity. Thus, the rotation matrix is a function of
two of the three Milankovitch parameters and is a valuable
and useful instructional tool/concept for lessons in geometry,
Geosci. Model Dev., 7, 1051–1068, 2014
T. S. Kostadinov and R. Gilb: Earth Orbit v2.1
mathematics, astronomy, physical geography, and climatology. At this point the Earth is correctly oriented in 3-D space
with respect to its orbit and the distant stars. Earth is then
translated to its proper instantaneous position on its orbit by
addition of its radius vector to all relevant Earth-bound model
elements (which are then plotted in the main heliocentric coordinate system).
Declination is one of the two spherical coordinates of the
equatorial astronomical coordinate system. It is measured
along a celestial meridian (hour circle) and is defined as the
angle between the celestial Equator and the direction toward
the celestial object (Meeus, 1998). Solar declination varies
with the seasons, due to obliquity. It is zero at the equinoxes,
reaches a maximum of +ε at summer solstice and a minimum of −ε at winter solstice. Solar declination determines
the length of day and the daily path of the Sun in the sky at
a given latitude (i.e., its altitude and azimuth above the horizon as a function of time). Thus, solar declination determines
instantaneous and time-integrated insolation. In turn, solar
declination and its evolution over the course of a year are a
function of the orbital elements; thus it provides the mathematical and conceptual link between the Milankovitch orbital
elements and insolation and climate. Here, we compute instantaneous solar declination using the angle between the direction of the North Pole and Earth’s radius vector, calculated
using their dot product. Thus, we explicitly compute solar
declination from the geometry of the model and it is a model
emergent property rather than prescribed a priori; therefore,
this also applies to insolation computations (Sects. 2.3 and
5).
2.2
Implementation of Kepler’s second law of
planetary motion
The heliocentric position of a planet in an elliptical orbit at
a given instant of time is given in terms of its true anomaly,
ν – see Eq. (1) and Sect. 2.1 above. True anomaly can also
be thought of as the angle (subtended at the Sun) which the
planet has “swept” from its orbit since last perihelion passage. Kepler’s second law of planetary motion states that the
planet will “sweep” equal areas of its orbit in equal intervals
of time and governs the value of true anomaly as a function
of time (e.g., Meeus, 1998; Joussaume and Braconnot, 1997).
At non-zero eccentricity, ν is not simply proportional to time
since last perihelion passage (time of flight) expressed as a
fraction of the orbital period in angular units. The latter quantity is called mean anomaly, M. Kepler’s second law is used
to relate M and ν, using an auxiliary quantity called eccentric anomaly, E. E and M are related by Kepler’s equation
(Meeus, 1998; Chapter 30):
E = M + e sin E,
(2)
where e is orbital eccentricity. When E is known, ν can be
solved for using (Meeus, 1998; Chapter 30):
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T. S. Kostadinov and R. Gilb: Earth Orbit v2.1
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Figure 2. (A) Present (J2000) orbital configuration for 16 September, using the La2004 solution and the calendar start date fixed at vernal
equinox on 20 March. The orbital ellipse is shown in blue, the semi-major and semi-minor axes (perpendicular to each other) are in red
and the lines connecting the solstices and equinoxes (also perpendicular to each other) are shown in black. The perihelion point, as well as
the equinoxes and solstices are labeled (NH = Northern Hemisphere). The Sun is shown as a semi-transparent yellowish sphere centered at
one of the orbital ellipse’s foci, both of which are marked with an “x” along the semi-major axis. The Earth is plotted with its center on
the corresponding place along the orbit, and the angle it has swept since last perihelion passage (the true anomaly angle) is filled in semitransparent light green. Earth’s Equator is plotted as a solid black line, and its axis of rotation is plotted as a dotted black line , with the North
Pole marked. The spheres of the Earth and the Sun are not to scale, the rest of the figure is geometrically/astronomically accurate and to scale.
This plot is in 3-D and has pan–tilt–zoom capability in the Earth Orbit v2.1 model. The corresponding GUI with numerical ancillary output
is shown in Fig. 1 (for latitude 43◦ N and So = 1366 W m−2 ). (B) Real orbital configuration for 16 September, 10 kyr in the future, using the
La2004 solution and a 20 March equinox as calendar start date. (C) Demo (imaginary) orbital configuration for 1 July (vernal equinox fixed
at 20 March), eccentricity = 0.6, obliquity = 45◦ , longitude of perihelion = 225◦ . The geometry is consistent with Berger et al. (2010), their
Fig. 1, although it is being viewed in (A) and (C) from the direction of fall equinox, as opposed to from the direction of vernal equinox in
their figure. The apparent eccentricity of the three orbits in Fig. 2 is also due to the view angle of the 3-D plot and the respective projection
onto a 2-D monitor/paper; the intrinsic eccentricity can be judged by tilting the plot or observing the relative distance from the two foci (the
Sun being at one of them) to the center of the ellipse, the intersection of the semi-major and semi-minor axes (red lines).
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1058
tan
T. S. Kostadinov and R. Gilb: Earth Orbit v2.1
ν
=
2
r
1+e
E
tan .
1−e
2
(3)
The forward Kepler problem consists of solving for time
of flight, M, given the planetary position, ν. This is
straightforward by first solving for E in Eq. (3) and using
it to solve for M in Eq. (2).
However, in the most intuitive case, which is implemented
here, the user enters a desired date, and the position of the
planet has to be determined from the date (i.e., time of
flight/mean anomaly M is given, and true anomaly has to be
determined). This is referred to as the inverse Kepler problem and amounts to solving for E in Eq. (2) and then for ν
in Eq. (3). Solving for E is not straightforward, as no analytical solution exists. Numerous numerical methods exist
for the solution of the inverse Kepler problem. Here, the binary search algorithm of Sinnott (1985) is used, as given in
Meeus (1998). It has the advantage of being computationally
efficient, which becomes important when time series of insolation is the desired model output. It also has the distinct
advantages of being valid for any value of eccentricity and
converging to the exact solution to within the user machine’s
precision.
2.3
Implementation of Insolation Computation
Instantaneous insolation at the top of the atmosphere (TOA)
can be computed as
r 2
o
sin h,
(4)
S(h, r) = So
r
where r is the length of the radius vector of Earth expressed
in AU, and h is the altitude of the Sun above the horizon
(e.g., Berger et al., 2010). Equation (4) is an expression of the
inverse square law and the Lambert cosine law of irradiance.
The radius vector length is computed in the model for the
chosen date (and not for every instant) using Eq. (1). So is
the TSI at ro = 1 AU by definition (Sect. 2). In this equation
insolation, S, is defined as the total (spectrally integrated)
solar radiant energy impinging at the TOA on a unit surface
area parallel to the mathematical horizon at a given latitude at
a given instant. S carries the units of So , here W m−2 . S needs
to be integrated over time and/or space in order to compute
insolation quantities of interest. Here, the main discrete time
step over which S is computed and output is one 24 h period
(i.e., daily insolation).
Daily insolation is a function of latitude, date, and So . The
date is associated with a given true anomaly for a given calendar start date and orbital configuration (Joussaume and Braconnot, 1997; Sect. 2.3.1). This determines the current solar
declination and the length of the radius vector of Earth (i.e.,
the Sun–Earth distance). The user inputs the desired latitude,
date and TSI, and the rest of the quantities are computed from
Geosci. Model Dev., 7, 1051–1068, 2014
the model geometry. Solar declination and the latitude determine the daily evolution of solar altitude, h, as a function of
time, as follows (e.g., Meeus, 1998):
sin h = sin δ sin ϕ + cos δ cos ϕ cos t .
(5)
In the above equation δ is solar declination, ϕ is geographic
latitude on Earth, and t is the hour angle of the Sun. δ is
assumed constant for the day of interest, and t is a measure of
the progress of time (e.g. Berger et al., 2010). Note that this
assumes the time derivative of the solar hour angle is equal to
one, i.e. it ignores the time derivative of the Equation of Time
(or equivalently, the annual variability in the time derivative
of the right ascension of the Sun is ignored). Half the day
length, ts , (i.e., the time between local solar noon and sunset),
is determined by setting h = 0◦ in Eq. (5):
ts = arccos(− tan ϕ tan δ).
(6)
In Eq. (6) ts is expressed in terms of hour angle of the Sun in
angular units. Equation (5) is integrated over time (under the
assumptions here, equivalently, over hour angle) from solar
noon to sunset in order to compute the time-average of the
sine of the solar altitude for the given date and latitude:
1
sin h =
ts
Zts
(sin δ sin ϕ + cos δ cos ϕ cos t)dt.
(7)
0
Equation (7) is integrated numerically with a very small time
step of about 10 s. Because the altitude of the Sun is symmetric about solar noon, it is sufficient to integrate only from
solar noon to sunset time. Daily insolation is then computed
by using the time-averaged sin h quantity in Eq. (4). The results are scaled by multiplying by the actual day length and
dividing by 24 h. The resulting quantity represents the mean
daily insolation over a full day, which is the standard value
used in climate and paleoclimate science (e.g., Laskar, 2014).
If this daily insolation is multiplied by 24 h (in seconds), total
energy receipt for that day (in J m−2 ) can be calculated.
At high latitudes, there are periods of the year with no sunset or no sunrise. These cases depend on the relationship of
latitude and solar declination (e.g., Berger et al., 2010). They
are handled separately by either integrating Eq. (7) over 24 h,
or, in the case of no sunrise (polar night), assigning a value
of exactly 0 W m−2 to daily insolation.
2.3.1
Integrating insolation over longer time periods –
caveats
Because of the varying eccentricity and longitude of perihelion, there is no fixed correspondence between true anomaly
(or true longitude) and any one single calendar date, even if
one were to define a fixed calendar start date. True anomaly
and longitude are the astronomically rigorous ways to define
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T. S. Kostadinov and R. Gilb: Earth Orbit v2.1
a certain moment in Earth’s year and seasons (e.g., Berger
et al., 2010). If one wishes to make insolation comparisons
between different orbital configurations, one must strictly
define a calendar start date, and even then insolation will
be in phase for different geological periods only for that
date (Joussaume and Braconnot, 1997). Thus, the question
of “what is insolation on June 20” is ill posed, unless one
defines strictly what is meant by the date of 20 June. The
problem persists if one wishes to compare insolation integrated over periods of time longer than a day, because over
geologic timescales, absolute values and the interval of true
longitudes “swept” between two classical calendar dates are
not constant. Thus, there are two ways to define a calendar
– the classical or fixed-day calendar, in which month lengths
follow the present-day configuration and the date of vernal
equinox is fixed, or a fixed-angular calendar, which defines
months beginning at certain true longitudes (function of true
anomaly and the precession phase, see also Sect. 2.1) and
they can therefore have a different number of days depending on the orbital configuration (Joussaume and Braconnot,
1997; Chen et al., 2011). The time intervals between solstices
and equinoxes also varies, because of varying eccentricity
and because these intervals happen in different places in the
orbit with respect to perihelion. Thus season lengths vary
over geologic time. Earth Orbit v2.1 outputs season length
in the main GUI to emphasize this important fact (Fig. 1).
Earth Orbit v2.1 uses the classical calendar dates (24 h periods) as the user time input, rather than true anomaly or true
longitude. This choice is much more intuitive to non-experts,
and best serves the educational purposes of the model. The
user has as a choice of calendar start date (Sect. 3) and true
solar longitude is output (Sect. 4.2; Fig. 1) to remind users
of the above considerations. The effect of calendar choice
on insolation phases and comparisons and on climate models
is discussed at length by Joussaume and Braconnot (1997),
Timm et al. (2008) and Chen et al. (2011).
The time step of integration can also influence the results
of insolation computations, for example, if annual insolation
is averaged with a 5-day step, results are substantially different from the case when a 1-day step is used (not shown).
For this reason, the model computes annually averaged insolation at a given latitude by using 1-day steps of integration. Finally, we note that the daily insolation computations
of the model are robust and validated for real values of the orbital parameters (Sect. 5.1, but see also Sect. 6); however, the
model currently has limited functionality for making comparisons of insolation integrated over longer time periods
over different geologic scales. In order to make such comparisons, the use of the elliptical integrals method of Berger et
al. (2010) is recommended, as well as the Laskar et al. (2004)
methods, both of which come with accompanying software
(Berger, 2014 and Laskar, 2014). In addition, users are referred to the latest version of the AnalySeries software package (Paillard et al., 1996; Paillard, 2014) for additional insolation and time-series options. All of the above can also
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1059
be used for verification of the output of the model presented
here.
3
Model user interface
The Earth orbit model is provided as Supplement (see section Code availability & license). The model is developed
and runs in MATLAB® . All model control is realized via
a single, user-friendly GUI panel (Fig. 1). Users are presented with a choice between the Be78 and the La2004
astronomical solutions for eccentricity, obliquity and precession. A “demo” mode is also available. If a real astronomical
solution is chosen, users are asked to input a year before or
after present (defined as J2000, i.e., 1 January, 2000 at 12:00
noon UT, see Introduction) for which they wish to run the
model. The GUI only allows users to choose years within
the respective solution’s validity: the Be78 solution is available for 1000 kyr before and after present (J2000), whereas
the La2004 solution is available for 101 000 kyr in the past
and 21 000 kyr in the future. The La2004 solutions are provided by Laskar (2014) (specifically at http://www.imcce.fr/
Equipes/ASD/insola/earth/La2004/index.html) in tabulated
form in 1 kyr intervals. The Be78 solutions are obtained
by transcribing code from NASA GISS (see Acknowledgements). The model looks up the values of eccentricity, obliquity and precession for the chosen year and solution (using
linear interpolation between tabulated years if necessary),
and these values are used in subsequent visualizations and
analyses. If the user chooses the “demo” mode, they select,
independently of each other, the values of the Milankovitch
parameters, which can be greatly exaggerated. In this way
users can isolate the effects of each parameter on orbital geometry, the seasons, and insolation. The “demo” mode is central to the pedagogical value and applications of the model
because it allows users to build and visualize an imaginary
orbit of, for example, very high eccentricity while keeping
obliquity fixed. Moreover, the model will output all subsequent parameters, such as solar declination, day length and
radius vector length, based on this exaggerated imaginary orbit.
Users input the desired calendar date, geographic latitude
on Earth (positive degrees in the Northern Hemisphere and
negative degrees in the Southern Hemisphere), and desired
value of TSI. The calendar date defaults to the current date,
latitude defaults to 43◦ N, and TSI defaults to 1366 W m−2
(Sect. 2). Two choices of calendar start date are available:
either fix vernal equinox to be at the beginning of 20 March
(default), or fix perihelion to be at the beginning of 3 January.
The availability of this choice complicates interpretation of
model output; however it has high instructional value. It illustrates that the choice of calendar start date and a calendar
system is a human construct, accepted by convention; it is
based on the actual year and day length but is relative. This
can also help test knowledge of the concepts explained in
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1060
T. S. Kostadinov and R. Gilb: Earth Orbit v2.1
Sect. 2.3.1. The effect of the different choice of calendar start
date is most apparent at exaggerated eccentricities and/or at
longitudes of perihelion that are very different from the contemporary value. Insolation time-series output (Sect. 4) is
only computed for the calendar being fixed to vernal equinox
on 20 March.
4
4.1
Model output
Graphical output
The main output of the model is a 3-D plot of Earth’s orbital
configuration. Figure 2a illustrates the orbital configuration
using the contemporary values of the Milankovitch parameters (the La2004 solution for J2000 is shown), for 16 September. The current phase of the precession cycle is such that
Northern Hemisphere (NH) winter solstice occurs shortly before perihelion (longitude of perihelion is ∼ 102.9◦ ). This
results in Northern Hemisphere spring and summer being
longer than the respective fall and winter (as shown in the
GUI, Fig. 1). Figure 2b illustrates the orbital configuration
also on 16 September, but for 10 kyr in the future (also using the La2004 solution). Since this represents about a half
of a precession cycle, the timing of the seasons is approximately 180◦ out of phase with respect to the contemporary
configuration (the longitude of perihelion is ∼ 279.2◦ , and
Northern Hemisphere summer occurs near perihelion and is
the shortest season). Because we chose to fix the calendar
start date such that vernal equinox is always on 20 March,
and the eccentricity is fairly low, the date 16 September still
occurs near the fall equinox, like in the contemporary example. However, because the length of time passing between
vernal equinox and fall equinox is now shorter, 16 September almost coincides with fall equinox, unlike the contemporary case. Of course obliquity and eccentricity have also
changed 10 kyr in the future, but unlike the longitude of perihelion, their changes are small in absolute terms, and thus
this cannot be readily visualized by comparing Fig. 2a and b.
This is one reason why it is very useful to have the ability to
choose arbitrary independent values of the Milankovitch parameters in the demo mode, constructing an imaginary orbit.
Figure 2c illustrates one example of such an imaginary orbit with greatly exaggerated eccentricity (0.6) and obliquity
(45◦ ) and longitude of perihelion of 225◦ , that is, very different from the J2000 values. This imaginary orbit illustrates
that the date 1 July can occur in the fall, due to the large eccentricity and the specific phase of precession chosen. Spring
lasts only ∼ 20 days in this configuration because it occurs
during perihelion passage, where the planet is much faster
according to Kepler’s second law, as compared to aphelion
passage (fall lasts ∼ 229 days in this configuration). Summer lasts about 58 days. Thus, July 1 occurs during the fall
season, counterintuitively. Importantly, such an exaggerated
eccentricity means that the planet is very close to the Sun
Geosci. Model Dev., 7, 1051–1068, 2014
during perihelion, and some really high insolation values can
occur even at modest solar declinations (e.g., for 29 March,
at 43◦ N, solar declination is ∼ 27◦ , day length is ∼ 16 h,
and daily insolation is 3307 W m−2 , far exceeding any contemporary value anywhere on Earth). The reason is that the
Sun–Earth distance then is only 0.4 AU, and the distance factor becomes a first-order effect on insolation, whereas it is a
second-order factor in the real Earth orbit configuration (angle being the first-order factor, see Eq. 4).
The plots of Fig. 2 have pan–tilt–zoom capability, so users
can view the orbital configuration from many perspectives;
this is at the core of the pedagogical value of the model.
The plot is updated with the current parameter selections by
pressing the “Plot/Update Orbit” button. Finally, note that the
apparent eccentricity of the orbits also changes with the view
angle and the projection onto a 2-D screen. This should not
be confused with the intrinsic orbital eccentricity, which can
be also judged by the relative distance of the orbital foci
(marked with an “x”) from the ellipse’s center (the intersection of the semi-major and semi-minor axes, red lines in
Fig. 2)
Users are presented with several options of plotting insolation as a function of time and latitude. First, insolation can
be plotted for a single year (using the currently selected Milankovitch parameters) as a function of day of year and latitude (Fig. 3a, upper panel). Insolation anomalies with respect
to the J2000 La2004 orbital configuration are also plotted,
using So = 1366 W m−2 (Fig. 3a, lower panel). Anomalies
are especially useful when analyzing the effect of changes
in insolation on the glacial–interglacial cycles. For example,
the anomalies at 65◦ N during summer months 115 kyr before
present (Fig. 3a, lower panel) suggest the inception of glaciation (e.g., Joussaume and Braconnot, 1997), as these areas
were receiving about 35–40 W m−2 less insolation than they
are receiving now. The data in these plots is computed with
steps of 5 days and 5 degrees of latitude. Multi-millennial insolation time series can also be plotted in a 3-D surface plot
as a function of year since J2000 and day of year, at the selected latitude. Users select the start and end years for the
time series. The data for these plots are computed for steps
of 1 kyr and one day (for day of year). An example of the
output is provided in Fig. 3b.
Several time-series line plots are also produced. Insolation
time series are plotted for the currently selected latitude; both
the currently selected date and the annual average are shown
(Fig. 4a). A multi-panel plot (Fig. 4b) allows comparison of
the three Milankovitch parameters. Precession is visualized
as the longitude of perihelion, as well as the climatic precession parameter, e sin ω (Berger and Loutre, 1994). A separate GUI button allows users to optionally produce timeseries plots of several paleoclimatic data sets (Fig. 4c). The
top panel shows the EPICA CO2 (Lüthi et al., 2008a, b) and
deuterium temperature (Jouzel et al., 2007a, b) time series
which go back to ∼ 800 kyr before present. The bottom panel
of Fig. 4c shows two benthic oxygen isotope (δ 18 O) data
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T. S. Kostadinov and R. Gilb: Earth Orbit v2.1
1061
Figure 3. (A) A day of year-latitude insolation plot for 115 kyr before present (J2000) (upper panel) and the corresponding anomaly from
J2000 (lower panel), using So = 1366 W m−2 . (B) Insolation time series at 65◦ N as a function of day of year, spanning 200 kyr before and
after present (J2000). Negative years are in the past. Both (A) and (B) use the La2004 solution.
set compilations – the Lisiecki and Raymo (2005) benthic
stack (Lisiecki, 2014) and the Zachos et al. (2001) data (Zachos et al., 2008). These data sets go back to 5320 kyr and
67 000 kyr before present, respectively. To first order, higher
δ 18 O values are associated with higher continental ice sheet
volumes and lower benthic ocean water temperatures (Zachos et al., 2001). For this reason, the y axis of the lower
panel of Fig. 4c is inverted, so that higher values of EPICA
CO2 and temperature (generally warmer climates) from the
upper panel of Fig. 4c can be easily associated with lower
δ 18 O values (also generally warmer climates). These paleoclimatic data are included for convenience of the user and
no further interpretation or analyses are provided. Users are
cautioned that the interpretation of these paleoclimatic signals and their uncertainties, time resolution and chronology
(age models) is fairly complex (e.g., Bradley, 2014, and data
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source references) and beyond the scope of this work. They
are provided here for illustrative purposes only, e.g., this enables users to easily visualize the last few glacial–interglacial
cycles (and the mid-Pleistocene transition to 100 kyr cyclicity, see Introduction), or to visually correlate these paleoclimatic time series with the corresponding Milankovitch parameter and insolation curves.
4.2
Numerical/Ancillary output
Ancillary data (and their units) are output in the main GUI
window (Fig. 1) and are updated every time the Earth orbit plot (Fig. 2) is re-drawn (Sect. 4.1) (i.e., every time the
“Plot/Update Orbit” button is pressed). Variables that are output in the main GUI are as follows: solar declination, insolation at the TOA for the chosen date and latitude, day length,
Geosci. Model Dev., 7, 1051–1068, 2014
1062
T. S. Kostadinov and R. Gilb: Earth Orbit v2.1
A)
Insolation at 65o latitude, W m−2
218
540
216
Insolation for June 20, W m −2
520
215
500
214
480
213
460
212
211
440
−200
dimensionless
B)
Annual mean of daily insolation, W m −2
217
−150
−100
−50
0
Thousand of years since J2000
50
100
150
200
Eccentricity
0.06
0.04
0.02
0
−500
−400
−300
−200
−100
0
100
200
300
400
500
100
200
300
400
500
Obliquity
25
degrees
24
23
, deg.
tilde
Long. of perihelion, ω
−400
−300
−200
−100
0
Longitude of perihelion/Climatic Precession
360
0.05
270
0.025
180
0
90
0
−500
−0.025
−400
−300
−200
−100
C)
0
100
200
300
−0.05
500
400
Climatic precession, e*sin( ω)
22
−500
EPICA CO 2 and temperature
5
280
−2.5
220
−5
2
[CO ], ppmv
0
240
Temperature, °C
2.5
260
−7.5
200
−10
180
−1000
−900
−800
−700
−600
−500
−400
Thousands of years since J2000
Lisiecki and Raymo 2005
δ18 O
−300
Zachos et al. (2001)
−200
−100
0
−200
−100
0
δ18 O
18
Benthic Foraminiferal δ O, ‰
3
3.5
4
4.5
5
−1000
−900
−800
−700
−600
−500
−400
Thousands of years since J2000
−300
Figure 4. (A) Insolation time-series plot spanning 200 kyr before and after present (J2000) at 65◦ N on 20 June (blue) and annual average
(red); (B) time-series plots of Milankovitch orbital parameters spanning 500 kyr before and after present. Panels from top to bottom display
eccentricity, obliquity, and longitude of perihelion and climatic precession; both (A) and (B) use the La2004 solution. (C) Time-series plots
of paleoclimatic data spanning one million years before present: EPICA ice core CO2 (blue) and deuterium temperature (green) (upper panel)
and the Lisiecki and Raymo (2005) (blue) and Zachos et al. (2001) (red) compilations of benthic oxygen isotope (δ 18 O) data (lower panel).
Note the y axis of the δ 18 O plot is inverted. Negative years for all Fig. 4 panels are in the past.
Geosci. Model Dev., 7, 1051–1068, 2014
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T. S. Kostadinov and R. Gilb: Earth Orbit v2.1
Sun–Earth distance, length of the seasons (as defined in the
North Hemisphere (NH)), the longitude of perigee, and true
and mean longitude of the Sun. As a reminder, the longitude of perigee is the angle between the directions of vernal
equinox and perihelion and true longitude is the angle Earth
has swept from its orbit, subtended at the Sun, since it was
last at vernal equinox; mean longitude is proportional to time
instead (for detailed definitions, see Sects. 2 and 2.1 above).
Users also are given the option of saving the data used to
make the insolation plots in Fig. 3 in ASCII format. The first
row and column of these files list the abscissa and ordinate
values of the data, respectively.
5
5.1
Model validation
Insolation validation
Daily insolation is the most important model output from climate science perspective and is the fundamental discrete time
unit at which the model calculates energy receipt at the TOA.
Daily insolation was validated against the results of Laskar et
al. (2004), as provided in Laskar (2014) (specifically, the precompiled Windows package at http://www.imcce.fr/Equipes/
ASD/insola/earth/binaries/index.html). In both the Earth orbit model and the Laskar software, the La2004 solution for
the orbital parameters was used, and the default model solar
constant (Table 1) was used. Laskar (2014) defines 21 March
as vernal equinox, whereas Earth Orbit v2.1 fixes vernal
equinox on 20 March for insolation time series. This was
taken into account in this validation. Two dates were tested
– 21 March and 20 June (according to the Earth Orbit v2.1
calendar; this corresponds to 1◦ and 90◦ mean longitude for
the Laskar (2014) software), at three latitudes −20◦ S, 45◦ N
and 65◦ N. The entire time series from 200 kyr in the past
to 200 kyr in the future (present = J2000) were tested with a
time step of 1 kyr. Validation is excellent; virtually all test
cases result in differences in insolation of less than 1 W m−2
for 21 March and less than 2 W m−2 for 20 June, respectively
(Fig. 5a and b, solid lines with dots), which corresponds to
less than 0.5 % of the absolute values (Fig. 5c and d, solid
lines with dots). Importantly, these differences are generally
much smaller or of the same order of magnitude as the corresponding differences between the Be78 and La2004 astronomical solutions as computed by Earth Orbit v2.1 (Fig. 5,
dashed lines). Furthermore, these differences are generally
smaller than the uncertainty resulting from varying estimates
of the TSI (e.g., Fröhlich, 2013, vs. Kopp and Lean, 2011,
see Sect. 2); also, these differences are smaller than the total contemporary anthropogenic radiative forcing on climate
due to fossil fuel emissions (IPCC, 2013; their Fig. SPM. 5).
The Earth Orbit v2.1 model uses its own internally constructed orbital geometry and first principles equations to
compute insolation. There is no additional a priori prescribed
constraint to the model other than the orbital elements
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astronomical solution and the semi-major axis and orbital
period (Sects. 2 and 2.3; Table 1). Therefore the validation
presented here is an independent verification of the model’s
geometry and computations, taking the Laskar (2014) values
as truth. Section 6 discusses sources of model uncertainty
which can explain some of the small differences observed.
5.2
Solar declination validation and season length
validation
Solar declination was validated against the algorithms of
Meeus (1998). The model year is neither leap, nor common
(Table 1) and is thus not equivalent to any single Gregorian
calendar year. In order to validate declination at all dates,
the Meeus (1998) algorithm was used to compute solar declinations for 12:00 UT on each date of four years (2009–
2012, 2012 being leap) and average the declinations for each
date (not day of year, Fig. 6). These averages were then
compared to the solar declination output by the model for
that date. Results indicate differences are always less than
∼ 0.2◦ (Fig. 6, black line). By construction, model solar declination on 20 March will always be exactly zero degrees.
In reality, the exact instance of vernal equinox varies year
to year, so these validation differences are expected. Importantly, the differences between the model and the 4-year averaged Meeus (1998) declinations are consistently smaller than
the daily rate of change of declination (Fig. 6, green curve),
as computed from the Meeus (1998) data. Additionally, these
differences are of a similar magnitude to the standard deviation of declination between these four years for each date
(Fig. 6, red curve). Thus the solar declination validation is
excellent and model configuration for each date is representative of a typical generic Gregorian calendar date. The discontinuities in the Meeus (1998) – derived curves in Fig. 6 (red
and green) are due to omitting 29 February 2012 when averaging declination values for each date. The discontinuities
in the Earth Orbit v2.1 to Meeus (1998) comparison curve
(Fig. 6, black curve) are due to the above, plus the fact that
the length of the model year is equal to the sidereal orbital period and thus 19 March is a longer “day” in the model year,
since calendar start is fixed as vernal equinox on 20 March
(also see Sect. 6 below). Finally, season lengths are an excellent method to validate the geometry of the model, because
they test that the model is correctly computing a given time
of flight on the orbit for a section of the orbit that corresponds
to a given season, and generally not coinciding with special points such as perihelion. Season lengths agree to within
0.01 days with the tabulated values of Meeus (1998) (his Table 27F).
Geosci. Model Dev., 7, 1051–1068, 2014
1064
T. S. Kostadinov and R. Gilb: Earth Orbit v2.1
A)
20S
EO v2.1 − La2004 (solid lines w/ dots) and Be78−La2004 (dashed lines) for March 21
45N
65N
20S
45N
65N
2
Wm
−2
1
0
−1
−2
−200
−100
−50
0
50
100
150
200
100
150
200
100
150
200
100
150
200
EO v2.1 − La2004 (solid lines w/ dots) and Be78−La2004 (dashed lines) for June 20
2
Wm
−2
B)
−150
0
−2
−200
C)
−150
−100
−50
0
50
EO v2.1 − La2004 % (solid lines w/ dots) and Be78−La2004 % (dashed lines) for March 21
1
Percent
0.5
0
−0.5
−1
−200
D)
−150
−100
−50
0
50
EO v2.1 − La2004 % (solid lines w/ dots) and Be78−La2004 % (dashed lines) for June 20
1
Percent
0.5
0
−0.5
−1
−200
−150
−100
−50
0
Years since J2000
50
Figure 5. Absolute differences (W m−2 , solid lines with dots) between our insolation solution (using the La2004 astronomical parameters)
and the Laskar (2014) insolation solution (also using the La2004 astronomical solutions; insolation provided by his Windows pre-compiled
package at http://www.imcce.fr/Equipes/ASD/insola/earth/binaries/index.html) for 21 March (A) and 20 June (B). Differences between the
Be78 and La2004 astronomical solutions (insolation for both computed by our model) are shown for comparison with dotted lines. Data
are shown for three different latitudes – 20◦ S (red), 45◦ N (green), and 65◦ N (blue). (C) same as in (A) but displaying percent insolation
difference, (D) same as in (B) but displaying percent insolation difference. Earth Orbit v2.1 insolation computations use the model’s own
orbital geometry with no additional a priori input other than the Milankovitch parameter solutions of La2004. Negative years are in the past.
See Sect. 5.1 for details.
6
Sources of uncertainties
Assumptions and approximations in the model and the underlying astronomical solutions propagate to uncertainties in
the model outputs, such as declination and insolation. Some
of these assumptions were already discussed, such as calculating insolation for a given calendar date vs. true longitude
interval (fixed-date vs. fixed-angle calendars), and choosing
integration steps for insolation time series (Sect. 2.3.1). The
calendar bias discussed in detail in Sect. 2.3.1 means that
if one compares insolation over geologic time on a given
classical calendar date (e.g., 16 September), which occurs a
given number of 24 h periods after the fixed vernal equinox,
one is not necessarily comparing insolation at the same true
longitude. The same argument is valid for an arbitrary interval of time longer than a day and shorter than a full orbital
cycle. This calendar bias creates the artificial north–south tilt
observed in insolation anomalies (Chen et al., 2011), which is
also exhibited by the Earth Orbit v2.1 model output (Fig. 3a,
second panel). This is expected because Earth Orbit v2.1 uses
the classical calendar dates, which are more user friendly.
Geosci. Model Dev., 7, 1051–1068, 2014
Next, we draw the users’ attention to a few additional
sources of uncertainty. Determination of some of these uncertainties is outside the scope of this work; however, users can
run sensitivity analyses using the model in order to quantify
them. Importantly, uncertainties in the astronomical solutions
that are used as input to the model will propagate to insolation computations. There are differences between the different astronomical solutions (e.g., Fig. 5). Accuracy is highest
near the present time and degrades further into the past or future (Laskar, 1999; Laskar et al., 2004). Chaotic components
of planetary orbital motions introduce an uncertainty that increases by an order of magnitude every ten million years,
making it impossible to obtain astronomical solutions for the
Milankovitch parameters over period longer than a few tens
of millions of years (Laskar et al., 2004). As a reminder, the
Be78 solution is valid for one million years in the past or future, whereas the La2004 solution is valid from 101 million
years before present to 21 million years in the future; however, solutions for times further back in time than 50 million
www.geosci-model-dev.net/7/1051/2014/
T. S. Kostadinov and R. Gilb: Earth Orbit v2.1
δ
0.4
EO
,°
dδM/dt, °/day
0.3
Degrees or Degrees/Day
1065
σδ , °
M
0.2
0.1
0
−0.1
−0.2
−0.3
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Date
Figure 6. Solar declination validation: difference between solar
declination as computed by the internal geometry of the Earth orbit model (using the La2004 orbital parameters for J2000) and
mean actual declination from the years 2009, 2010, 2011 and 2012
as computed for 12:00:00 UT every day with the algorithms in
Meeus (1998) (black solid line). The rate of change of declination (green solid line) and the standard deviation of declination for
each date for the four years (red solid line, N = 4 for each data
point) are also shown for reference. The model computations were
performed with the calendar start date fixed at vernal equinox of
20 March. The date 29 February 2012 was removed from the analysis, so the abscissa corresponds to a given date, that is, dates, not
days of year were averaged for a given mean solar declination across
the four years. Abscissa ticks represent the 15th of each month. If
00:00 UT is used for the Meeus computations instead, differences
(black curve) have a different pattern and are larger, but never exceed ∼ 0.4 degrees (not shown). See Sect. 5.2 for details.
years before present should be treated with caution (Laskar
et al., 2004).
Due to the gravitational interaction of Earth and other solar system bodies, in particular Jupiter, Venus and the Moon,
high-frequency variability (timescales of years to centuries)
of the Milankovitch parameters is superimposed on the longterm low-frequency Milankovitch cycles. An example of
such variability is the nutation in obliquity with a period of
∼ 18 years. These high-frequency fluctuations also lead to
insolation changes. Bertrand et al. (2002) used results from
the VSOP82 planetary position solution (Bretagnon, 1982)
and a simple climate model to demonstrate that the amplitudes of these high-frequency variations and the effect on insolation and surface temperature is negligible (equivalent to
model noise) as compared to the 11-year Sun cycle or the
low-frequency trends.
The model prescribes the sidereal year as the orbital period (Table 1), which is slightly longer than the tropical year
(Meeus, 1998). The difference is on the order of 0.01 days.
The use of these two different period definitions leads to negligible differences in solar declination on a given date (for the
www.geosci-model-dev.net/7/1051/2014/
J2000 La2004 orbital parameters; not shown), much smaller
than the validation differences of Fig. 6. We conclude that
the choice of orbital period does not influence the insolation
computations significantly.
A single value for solar declination and the radius vector length is used in the computation of daily insolation
(Sect. 2.3). In reality, these quantities change continuously,
instead of having discrete values. This is likely to introduce
small errors in insolation that will generally be smaller in
magnitude than the difference in daily insolation between
successive days. Importantly, day length and daily insolation
values near perihelion at very high eccentricities (that can occur only in Demo mode, not in the real Milankovitch cycles)
should be treated with caution due to significant violation of
the assumptions in applying Eqs. (6) and (7) (see Sect. 2.3).
In such cases the radius vector length and the declination of
the Sun may change significantly over the course of one 24 h
period, and the hour angle of the Sun changes significantly
more slowly than assumed (its time derivative is less than
one); this is not dealt with rigorously in this implementation.
Sunrise and sunset times used in the insolation computation are referred to the center of the disk of the Sun and the
mathematical horizon at the given latitude. Note also that irradiance is given at the top of the atmosphere (TOA), but all
computations are geocentric, rather than topocentric, which
should lead to negligible insolation differences. Since the
model year is not an integral number of days, if total annual
insolation is computed by summing daily insolation values,
the 19 March insolation needs to be scaled by 1.256363 to
reflect the fact that this day is 24 × 1.256363 h long in the
model (Berger et al., 2010). Here, we average daily insolation to output average annual insolation, so this correction is
not applied.
7
Concluding remarks
We presented Earth Orbit v2.1, an interactive 3-D analysis and visualization model of the Earth orbit, Milankovitch
cycles, and insolation. The model is written and runs in
MATLAB® and is controlled from a single integrated userfriendly GUI. Users choose a real astronomical solution for
the Milankovitch parameters or user-selected demo values.
The model outputs a 3-D plot of Earth’s orbital configuration (with pan–tilt–zoom capability), selected insolation
time series, and numerical ancillary data. The model is intended for both research and educational use. We emphasize
the pedagogical value of the model and envision some of
its primary uses will be in the classroom. The user-friendly
GUI makes the model very accessible to non-programmers.
It is also accessible to non-experts and the primary and
secondary education classroom, as minimal scientific background is required to use the model in an instructional setting. Disciplines for which the model can be used span mathematics (e.g., spherical geometry, linear algebra, curve and
Geosci. Model Dev., 7, 1051–1068, 2014
1066
surface parameterizations), astronomy, computer science, geology, Earth system science, climatology and paleoclimatology, physical geography and related fields.
The authors encourage feedback and request that comments, suggestions, and reports of errors/omissions be directed to [email protected].
Code availability & license
The files necessary to run the model “Earth Orbit v2.1” in
MATLAB® are provided here as Supplement. In addition,
model files are expected to be available on the website of the
University of Richmond Department of Geography and the
Environment (http://geography.richmond.edu), under the Resources category; documented updates may be posted there.
Sources of external data files are properly acknowledged in
the file header and/or the ReadMe.txt file, as well as in this
manuscript. The GUI is raised by typing the name of the
associated script (“Earth_orbit_v2_1”) on the MATLAB®
command line. The model has been tested in MATLAB® release R2013b on 64 bit Windows 7 Enterprise SP1 and Linux
Ubuntu 12.04 LTS, but should run correctly in earlier versions of MATLAB® and on different platforms. The model
is distributed under the Creative Commons BY-NC-SA 3.0 license. It is free for use, distribution and modification for noncommercial purposes. Details are provided in the ReadMe.txt
file.
The Supplement related to this article is available online
at doi:10.5194/gmd-7-1051-2014-supplement.
Acknowledgements. Funding for this work was provided to
T. S. Kostadinov by the Andrew W. Mellon Foundation/Associated
Colleges of the South Environmental Postdoctoral Fellowship
and the University of Richmond School of Arts & Sciences. The
University of Richmond School of Arts & Sciences also provided
funding for R. Gilb. We would especially like to acknowledge Martin Medina for his help and the incredibly inspirational discussions.
This project would not have been conceived without him. We also
thank André Berger and Jacques Laskar for their Milankovitch and
insolation solutions and software, Jacques Laskar for providing
input data files for his solutions, and Gary L. Russell/NASA GISS
for the Berger Milankovitch solutions FORTRAN® code. We thank
David Kitchen, Aisling Dolan, Jon Giorgini and William Folkner
for the useful discussions. Oxygen isotope data compilations (LR04
and Zachos et al., 2001) and EPICA data sets authors and contributors are also hereby acknowledged. T. S. Kostadinov expresses his
gratitude to Van Nall and Della Dumbaugh for their multivariate
calculus and linear algebra classes at University of Richmond, as
well as to other mathematics and computer science faculty at the
University of Richmond. We thank Jean Meeus for his excellent
Astronomical Algorithms. T. S. Kostadinov thanks his astronomy
teacher, Vanya Angelova, for being instrumental in developing his
Geosci. Model Dev., 7, 1051–1068, 2014
T. S. Kostadinov and R. Gilb: Earth Orbit v2.1
interests in astronomy. We also thank the Wikipedia® Project and
its contributors for providing multiple articles on mathematics, astronomy, orbital mechanics, and Earth Science that are a great first
step in conducting research. We thank two anonymous reviewers
for providing constructive comments that improved this manuscript.
Edited by: D. Lunt
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